Abstract

A nonlinear fiber with a gain that has symmetric lateral frequency bands is found to generate a stable train of bright or dark pulses with a well-defined repetition rate. The pulse train is generated from noise; the pulses are unchirped, and their amplitude and repetition rate are determined by the parameters of the system. The underlying mechanism for this pulse formation is recognized as dissipative four-wave mixing in which only two frequencies are nonnegligible; one near the maximum gain transmits energy by wave mixing to its third harmonic, which is in the region of negative gain. An implicit analytical expression for the pulse train solution is derived. It is demonstrated that dissipative four-wave mixing can be employed to yield passive mode locking in a fiber ring laser consisting of only a filter and an active fiber. The resulting train of pulses is shown to have a high soliton content.

© 1998 Optical Society of America

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References

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  1. J. G. Caputo, N. Flytzanis, and M. P. Sørensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B 12, 139 (1995).
    [CrossRef]
  2. H. T. Moon, P. Huerre, and L. G. Redekopp, “Three frequency motion and chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 49, 458 (1982).
    [CrossRef]
  3. K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171 (1983).
    [CrossRef]
  4. W. Schopf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg–Landau equation for subcritical bifurcation,” Phys. Rev. Lett. 66, 2316 (1991).
    [CrossRef]
  5. D. Anderson and M. Lisak, “Modulation instability of coherent optical fiber transmission signals,” Opt. Lett. 9, 468 (1984).
    [CrossRef] [PubMed]
  6. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).
  7. P. A. Bélanger, L. Gagnon, and C. Paré, “Solitary pulses in an amplified, nonlinear, dispersive medium,” Opt. Lett. 14, 943 (1989).
    [CrossRef]
  8. Y. Chen, “Solitary waves in nonlinear media with gain and loss,” Electron. Lett. 27, 1985 (1991).
    [CrossRef]
  9. V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklarov, “Evolution of light in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530 (1988).
  10. M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
    [CrossRef] [PubMed]
  11. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fiber,” Opt. Lett. 14, 1008 (1989).
    [CrossRef] [PubMed]
  12. S. V. Chernikov, J. R. Taylor, P. V. Mamyshev, and E. M. Dianov, “Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers,” Electron. Lett. 28, 931 (1992).
    [CrossRef]
  13. G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252 (1992).
    [CrossRef]

1995 (1)

1994 (1)

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

1992 (2)

S. V. Chernikov, J. R. Taylor, P. V. Mamyshev, and E. M. Dianov, “Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers,” Electron. Lett. 28, 931 (1992).
[CrossRef]

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252 (1992).
[CrossRef]

1991 (2)

W. Schopf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg–Landau equation for subcritical bifurcation,” Phys. Rev. Lett. 66, 2316 (1991).
[CrossRef]

Y. Chen, “Solitary waves in nonlinear media with gain and loss,” Electron. Lett. 27, 1985 (1991).
[CrossRef]

1989 (2)

1988 (1)

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklarov, “Evolution of light in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530 (1988).

1984 (1)

1983 (1)

K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171 (1983).
[CrossRef]

1982 (1)

H. T. Moon, P. Huerre, and L. G. Redekopp, “Three frequency motion and chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 49, 458 (1982).
[CrossRef]

Anderson, D.

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

D. Anderson and M. Lisak, “Modulation instability of coherent optical fiber transmission signals,” Opt. Lett. 9, 468 (1984).
[CrossRef] [PubMed]

Bekki, N.

K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171 (1983).
[CrossRef]

Bélanger, P. A.

Bofetta, G.

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252 (1992).
[CrossRef]

Caputo, J. G.

Chen, Y.

Y. Chen, “Solitary waves in nonlinear media with gain and loss,” Electron. Lett. 27, 1985 (1991).
[CrossRef]

Chernikov, S. V.

S. V. Chernikov, J. R. Taylor, P. V. Mamyshev, and E. M. Dianov, “Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers,” Electron. Lett. 28, 931 (1992).
[CrossRef]

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fiber,” Opt. Lett. 14, 1008 (1989).
[CrossRef] [PubMed]

Dianov, E. M.

S. V. Chernikov, J. R. Taylor, P. V. Mamyshev, and E. M. Dianov, “Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers,” Electron. Lett. 28, 931 (1992).
[CrossRef]

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fiber,” Opt. Lett. 14, 1008 (1989).
[CrossRef] [PubMed]

Flytzanis, N.

Gagnon, L.

Grigoryan, V. S.

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklarov, “Evolution of light in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530 (1988).

Huerre, P.

H. T. Moon, P. Huerre, and L. G. Redekopp, “Three frequency motion and chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 49, 458 (1982).
[CrossRef]

Korytin, A. I.

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

Kramer, L.

W. Schopf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg–Landau equation for subcritical bifurcation,” Phys. Rev. Lett. 66, 2316 (1991).
[CrossRef]

Lisak, M.

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

D. Anderson and M. Lisak, “Modulation instability of coherent optical fiber transmission signals,” Opt. Lett. 9, 468 (1984).
[CrossRef] [PubMed]

Maimistov, A. I.

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklarov, “Evolution of light in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530 (1988).

Mamyshev, P. V.

S. V. Chernikov, J. R. Taylor, P. V. Mamyshev, and E. M. Dianov, “Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers,” Electron. Lett. 28, 931 (1992).
[CrossRef]

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fiber,” Opt. Lett. 14, 1008 (1989).
[CrossRef] [PubMed]

Moon, H. T.

H. T. Moon, P. Huerre, and L. G. Redekopp, “Three frequency motion and chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 49, 458 (1982).
[CrossRef]

Nozaki, K.

K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171 (1983).
[CrossRef]

Osborne, A. R.

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252 (1992).
[CrossRef]

Paré, C.

Prokhorov, A. M.

Redekopp, L. G.

H. T. Moon, P. Huerre, and L. G. Redekopp, “Three frequency motion and chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 49, 458 (1982).
[CrossRef]

Schopf, W.

W. Schopf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg–Landau equation for subcritical bifurcation,” Phys. Rev. Lett. 66, 2316 (1991).
[CrossRef]

Sergeev, A. M.

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

Sklarov, Yu. M.

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklarov, “Evolution of light in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530 (1988).

Sørensen, M. P.

Taylor, J. R.

S. V. Chernikov, J. R. Taylor, P. V. Mamyshev, and E. M. Dianov, “Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers,” Electron. Lett. 28, 931 (1992).
[CrossRef]

Vanin, E. V.

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

Vázquez, L.

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

Electron. Lett. (2)

Y. Chen, “Solitary waves in nonlinear media with gain and loss,” Electron. Lett. 27, 1985 (1991).
[CrossRef]

S. V. Chernikov, J. R. Taylor, P. V. Mamyshev, and E. M. Dianov, “Generation of soliton pulse train in optical fibre using two cw singlemode diode lasers,” Electron. Lett. 28, 931 (1992).
[CrossRef]

J. Comput. Phys. (1)

G. Bofetta and A. R. Osborne, “Computation of the direct scattering transform for the nonlinear Schroedinger equation,” J. Comput. Phys. 102, 252 (1992).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (3)

Phys. Rev. A (1)

M. Lisak, E. V. Vanin, A. I. Korytin, A. M. Sergeev, D. Anderson, and L. Vázquez, “Dissipative optical solitons,” Phys. Rev. A 49, 2806 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

H. T. Moon, P. Huerre, and L. G. Redekopp, “Three frequency motion and chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 49, 458 (1982).
[CrossRef]

K. Nozaki and N. Bekki, “Pattern selection and spatiotemporal transition to chaos in the Ginzburg–Landau equation,” Phys. Rev. Lett. 51, 2171 (1983).
[CrossRef]

W. Schopf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg–Landau equation for subcritical bifurcation,” Phys. Rev. Lett. 66, 2316 (1991).
[CrossRef]

Sov. Phys. JETP (1)

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklarov, “Evolution of light in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530 (1988).

Other (1)

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).

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Figures (7)

Fig. 1
Fig. 1

Aspect of the spectral gain, γ(ω) for αg=βg=0.1 and γ=0.05. The frequency for the maximum gain is ωm; the cut frequency is ωc.

Fig. 2
Fig. 2

Evolution of the number of quanta N for two values of the time slot T. The solid curve indicates the results for T=8; the dashed–dotted curve represents the results for T=10.

Fig. 3
Fig. 3

Temporal distributions |u(t, zi)| for the different types of evolution: (a) SYM state (T=8), (b) irregular evolution (T=10), (c) ASYM state (T=20) with pulses moving to the right, (d) ASYM state (T=20) with pulses moving to the left. The values of the parameters are α=1.0, αg=βg=γ0=0.1, and κ=2.0.

Fig. 4
Fig. 4

Fourier transform u˜(ω) at z=1000 for the following states: (a) SYM state (T=8), (b) irregular evolution (T=10), (c) ASYM state (T=20) for pulse trains moving to the right, (d) ASYM state (T=20) for pulse trains moving to the left. The alternation between the situations depicted in (c) and (d) depends randomly on the initial conditions. This is the only case for which we have found a dependence on the initial data. This asymmetry dimishes for higher densities of modes. The thick curves indicate γ(ω). The values of the parameters are α=1.0, αg=βg=γ0=0.1, and κ=2.0.

Fig. 5
Fig. 5

(a) Discretization of N versus T. Circles indicate an SYM state; stars indicate an ASYM state. (b) Observed values of ω* as a function of T, the value of the frequency for the maximum gain ωm=1/2.

Fig. 6
Fig. 6

Passive mode locking in the case of a high density of modes. For each value of γ0 the field has propagated to z=800. Other parameters are T=100, α=1.0, αg=βg=0.1, and κ=2.0. (a) Spectrum of the field. The highest intensities are indicated by the darkest shades. (b) Field in the time domain for γ0=0.07. (c) Field in the time domain for γ0=0.119.

Fig. 7
Fig. 7

Amplitudes of the two frequency components for the SYM state found from the analytical expression (curves) and the partial differential equation Eq. (1) (filled circles). The parameters are α=1.0, αg=βg=0.1, and κ=2.0, and γ0 is varied for a time window T=8.

Tables (1)

Tables Icon

Table 1 Maximum Length of the Ring for Reproduction of Autowave Generation in the Continuous Model

Equations (21)

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iuz+(α+iαg)utt+iβgutttt+κ|u|2u=iγ0u,
γ(ω)=γ0+αgω2-βgω4.
ωc=αg+(αg2+4βgγ0)1/22βg1/2;
u=a1 exp(iω*t)+a-1 exp(-iω*t)+a3 exp(i3ω*t)+a-3 exp(-i3ω*t)
i da1dz-[αω*2+iγ(w*)]a1+κ[a1(3|a1|2+4|a3|2)
+2a3|a1|2+a12a3*+2a32a1*]=0,
i da3dz-[9αω*2+iγ(3w*)]a3+κ[a3(3|a3|2
+4|a1|2)+a1|a1|2+2a12a3*]=0,
ddz (|a1|2+|a3|2)=γ1|a1|2+γ3|a3|2.
a1=a10 exp(ikz),
a3=a30 exp(iφ)exp(ikz),
a102a302=-γ3γ1,
a302=-γ1/κ-γ3/γ1 sin φ+2 sin 2φ,
k=-5αw*2-12 (7+2 cos 2φ)(γ1-γ3)+(3γ1-γ3)-γ3/γ1 cos φ-γ3/γ1 sin φ+2 sin 2φ.
b4 exp(i4φ)+b3 exp(i3φ)
+b2 exp(i2φ)-b3* exp(iφ)-b4*=0,
b4=i 16αω*2γ1+γ3-2i(γ3/γ1+1),
b3=8αω*γ1+γ3 -γ3/γ1-i(3+γ3/γ1)-γ3/γ1,
b2=-2i(1+γ3/γ1).
iuz+αutt+κ|u|2u
=iγ0+αg24βgu-iLkαg24βg u+αgutt+βguttttδ(z-zk),

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